Automatic Variationally Stable Analysis for Finite Element Computations: Transient Convection-Diffusion Problems

We establish stable finite element (FE) approximations ofconvection-diffusion initial boundary value problems using the automaticvariationally stable finite element (AVS-FE) method. The transientconvection-diffusion problem leads to issues in classical FE methods as thedifferential operator can be considered singular perturbation in both space andtime. The unconditional stability of the AVS-FE method, regardless of theunderlying differential operator, allows us significant flexibility in theconstruction of FE approximations. We take two distinct approaches to the FEdiscretization of the convection-diffusion problem: i) considering a space-timeapproach in which the temporal discretization is established using finiteelements, and ii) a method of lines approach in which we employ the AVS-FEmethod in space whereas the temporal domain is discretized using thegeneralized-alpha method. In the generalized-alpha method, we discretize thetemporal domain into finite sized time-steps and adopt the generalized-alphamethod as time integrator. Then, we derive a corresponding norm for theobtained operator to guarantee the temporal stability of the method. We present numerical verifications for both approaches, including numericalasymptotic convergence studies highlighting optimal convergence properties.Furthermore, in the spirit of the discontinuous Petrov-Galerkin method byDemkowicz and Gopalakrishnan, the AVS-FE method also leads to readily availablea posteriori error estimates through a Riesz representer of the residual of theAVS-FE approximations. Hence, the norm of the resulting local restrictions ofthese estimates serve as error indicators in both space and time for which wepresent multiple numerical verifications adaptive strategies.